The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

Until the present time there is no exact solution to the contact problem for a rectangular plate on an elastic base with distribution properties. Practical analogues of this design are slab foundations widely used in construction. A lot of scientists have solved this problem in various ways. The methods of finite differences, B. N. Zhemochkin and power series do not distinguish a specific feature in contact stresses at the edges of the plate. The author of the paper has obtained an expansion of the Boussinesq solution for determining displacements of the elastic half-space surface in the form of a double series according to the Chebyshev polynomials of the first kind in a rectangular region. For the first time, such a representation for the symmetric part of the Boussinesq solution was obtained by V. I. Seimov and it has been applied to study symmetric vibrations of a rectangular stamp, taking into account inertial properties of the half-space. Using this expansion, the author gives a solution to the problem for a rectangular plate lying on an elastic half-space under the action of an arbitrarily applied concentrated force. In this case, the required displacements are specified in the form of a double row in the Chebyshev polynomials of the first kind. Contact stresses are also specified in the form of a double row according to the Chebyshev polynomials of the first kind with weight. In the integral equation of the contact problem integration over a rectangular region is performed while taking into account the orthogonality of the Chebyshev polynomials. In the resulting expression the coefficients are equal for the same products of the Chebyshev polynomials. The result is an infinite system of linear algebraic equations, which is solved by the amplification method. Thus the sought coefficients are found in the expansion for contact stresses.

The need to cut construction cost of residential and public buildings and provide them with a free and transformable planning structure during their operation cause interest in building wall systems with a large step of bearing walls. In order to reduce labor inputs and increase rate of construction in such building load-bearing system it is also necessary to maximize the use of large-sized prefabricated products and minimize consumption of in-situ concrete. In this case prefabricated products should be substituted according to the conditions of local (regional) construction industry base and volume of in-situ concrete must be sufficient to ensure a complete redistribution of internal forces between elements of the bearing system under load. As for the described bearing wall system of a multi-storey building the paper presents a flat precast solid floor formed by hollow-core slabs and monolithic crossbars supported by load-bearing walls. The hollow-core slabs supported at the ends on cast-in-place crossbars in the planes of bearing walls are arranged in dense groups between cast-in-place braced cross-beams. Dense contacts between overlapping elements are fixed by internal bonds. New data on distribution of forces in floor elements under the action of a vertical load have been obtained on the basis of full-scale tests and existing theoretical assumptions. It has been established that due to this load reactive thrust forces ensuring an operation of every hollow-core slab group in the floor as an effective solid plate supported along the contour have been originated in the floor plane along two main axes. Calculation of the reactive thrust forces makes it possible more accurately to assess a load-bearing capacity and rigidity of the precast solid floor and to increase a step of bearing walls up to 8 m and more while having hollow-core slabs with a thickness of 220 mm.

Using the example of vertical displacements, it is shown that by combining a solution to the problem of determining vertical displacements from the action of four identical concentrated forces symmetrically applied to an elastic half-space and two identical concentrated forces symmetrically applied to an elastic quarter-space, one can obtain a solution about the action of one force on 1/8 of the elastic space with free edges. To find vertical displacements in an elastic half-space, the  Boussinesq  solution  is  used,  and  vertical  displacements in an  elastic  quarter-space – an integral equation obtained by Ya. S. Uflyand to determine vertical displacements in the face of a homogeneous elastic isotropic quarter-space, for which a deformation modulus and Poisson’s ratio are constant. However, an integral equation of Ya. S. Uflyand is very inconvenient for practical use, therefore, in the paper, an approximate expression written in terms of elementary functions is proposed to find vertical displacements in the face of an elastic quarter-space from the action of a concentrated force. To obtain the latter, a special approximation method is used. The desired solution is also expressed in terms of elementary functions. In this case, an accurate calculation is obtained for an incompressible material with Poisson’s ratio 1/8 of the space n = 0.5. Since the solution is obtained in the case of a concentrated force acting on 1/8 of the elastic space, it is easy to find an expression for determining the vertical displacements of the edge of 1/8 of the elastic space from the action of any distributed load by integrating over the area of action of this load from the influence function, which is taken as required decision. Recommendations for improving the accuracy of calculations are offered. The described approach can also be used to determine the stress-strain of 1/8 of the space with both hingedly supported and free edges.

The paper considers a solution of contact problem for hinged supporting node of beam floor slab (coating). The main goal is to determine a stress state of the area where a plate rests on the wall. In this case, a number of problems are solved: construction of reactive pressure diagrams in the area of direct plate and wall contact, clarification of the calculated plate span, influence of contact zone size on a value of maximum bending moment in the middle of the plate, determination of contact area at various indices of flexibility and comparison of the obtained results with the known solution of rigid stamp and elastic quarter-plane interaction. The calculation has been carried out by the Zhemochkin method, its implementation for the given task corresponds to a mixed method of structural mechanics. As an illustration, the calculation has been performed on a concentrated load applied in the middle of the plate span. In the course of the study, it has been established that when a reinforced concrete slab rests on concrete and brick walls, the contact zone reduces itself to two Zhemochkin sections. When a flexibility index is decreased that corresponds to slab support on a less rigid base, the contact area is increased, and that, in its turn, has an influence on an increase of the calculated slab span and the bending moment in the middle of the slab. In the case of an absolutely rigid plate support (flexibility index is equal to zero), the contact stresses reach their maximum value at the point of plate edge contact and elastic quarter-plane. The nature of the diagram is confirmed by an analytical dependence of contact stress distribution obtained by Aleksandrov V. M. when solving a problem of pressing a rigid stamp into an edge of an elastic wedge.